Curiosities and counterexamples in smooth convex optimization

Abstract

Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy’s gradient curves, convergence of Newton’s flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka–Łojasiewicz inequality. All examples are planar. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved \(C^k\) convex compact sets in the plane, we provide a level set interpolation of a \(C^k\) smooth convex function where \(k\ge 2\) is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore, given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.